Posted
by
Unknown Lamer
on Wednesday March 02, 2011 @03:00PM
from the hypergeometric-overlords dept.

peter.hill.1980 writes "Wall Street's money making formulas need to be as explicit as possible for efficiency purposes. An old, existing and famous formula — binomial options pricing formula — has now been scrutinized for theoretical optimality in a forthcoming paper by Evangelos Georgiadis of MIT using Gosper's Algorithm, proving that no general explicit or closed form expression exists for pricing."

Indeed, I realize that this is/., and that/. doesn't have any editors, but this is pretty ridiculous. At least link to something that has some information if you can't be arsed to create an informative summary.

It's very simple. What part of "We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach" you didn't understand?

It's very simple. What part of "We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach" you didn't understand?

The vanilla part, of course. After all, why should it matter if the options come in vanilla or chocolate flavour?:-)

Isn't it just a bet on the variance of the underlying stock? Skimming over those models, they look like they're really all just different ways of mutating your choice of initial assumption about the distribution of possible futures.

It absolutely is gambling and any investor who denies this is either dishonest or not very well informed. I infer from your sarcastic quote marks that I'll enjoy Edward Chancellor's book -- sometime after I enjoy Taleb's, and Reich's, and...

The refined financial term is "speculation". It only becomes "gambling" in the business sense when you crater something.

While I sense a great deal of sarcasm in this thread, it is worth noting that speculation actually can result in positive return over long periods of time. Some people are very good trader/gamblers and their talents shouldn't be denied. But the problem comes when you have high leverage amplifying all the risks of the market and the business.

While I sense a great deal of sarcasm in this thread, it is worth noting that speculation actually can result in positive return over long periods of time. Some people are very good trader/gamblers and their talents shouldn't be denied. But the problem comes when you have high leverage amplifying all the risks of the market and the business.

Totally agree. I almost said the same thing, but just wanted to be sure not to say too much. I tend to go on a bit, you see.:-)

But since somebody is interested, I'll say that a very important distinction is knowing the odds and playing them smartly, versus guessing the odds skillfully/luckily. Over time, luck always runs out eventually. So says the Strong Law of Large Numbers.

High leverage where one borrows staggering amounts of money (something like 50 borrowed dollars to 1 dollar of assets was common in the recent real estate crash) will guarantee a crash even for very stable investments.

In other words, a "black swan" is sure to come by, eventually. So only fools construct systems in which one "black swan" will

Hedging - and, in a sense, a lot of the issues around option pricing - are the opposite of gambling. They're more like running a bookmaker's. The idea is to limit your risk exposure whilst making a nice profit from the actual gamblers.

While this is not my area of expertise I actually know of an example. Several years ago when jet fuel prices skyrocketed and all the airlines started charging fees on everything to stay afloat Southwest Airlines didn't even raise their fares because they had hedged [nytimes.com] long term fuel oil at ~$50 barrel. At the time they set up the hedge the median price was much lower so it meant they were paying more than any of the other airlines for their fuel but it also meant that when the price shot up to >$90 barrel t

But was there a mathematical proof? Or just 'Street common wisdom & anecdote, that no sure-fire formula exists?

The summary: "... proving that no general explicit or closed form expression exists for pricing."

There is something new here. I get your point that Wall Street will not change its ways because you and every other trader have already been assuming what this paper proves, and I don't doubt what you said one bit, but for the rest of us this is significant because formal, incontrovertible mat

Option [wikipedia.org], e.g. "I pay you $100 and you agree to (sell to me / buy from me) 1,000 shares of XYZ at a locked-in price of $50 apiece whenever I decide to exercise my option" (I may decide not to exercise it at all, and I may have a time limit)

Okay, then. In my opinion, the color of Steve Ballmer's socks is responsible for Scott Walker, Koch Industries, the explosion of BP's Maconda Well, the disinformation from "Curveball" that the Junior Bush administration chose to use as the pretext for Operation Iraqi Liberation, the increasing rate of unemployment in the United States as well as all the very-long-term-unemployed who no longer qualify to even be included in the "official" unemployment statistics, and, oh yeah, exceptions to H1-B law obtaine [arstechnica.com]

The basic form of the algorithm (according to *AA groups) is as such: $Max_Payable_Price times ($Total_World_Population - $Steenking_Pirates). *AA's obviously want to minimize the $Steenking_Pirates, especially the ones who simply don't listen to their music in the first place.

Many lawmakers agree with this, with the agreement being proportional to the money they receive from the *AA's.

And yes, I know that people who don't listen to music shouldn't need to pay, but I dare you to tell the RIAA that. It's eve

And what does this have to do with option pricing? It just proves that there is no closed form. From the quick little research I did on closed forms, all this means is that you can't use limits or integrals, which are used as solutions for a slew of real world problems.

It seems, after reading through the paper (to the extent my non-MIT mind understood things) that this is based upon a pricing model of European options [slashdot.org]. European options can only be exercise on the expiry date, American options can be exercised any time before that date.

I am a modelier (a fancy pants way of "I configure and run models") though not in Economics. This has *huge* implications for the practical application of models. Now, what no closed form solution means is that there may be a number of different paths that a solution can be achieved. You can converge from a number of different directions, and be "right for the wrong reasons". "Great!", you say, "if I am wrong on one of my parameters, this means

Although I've only looked at the paper briefly I think that what it proves is that there is no closed-form expression for the partial sums in the Cox-Ross-Rubenstein model. Such a closed form expression would be neat, but ultimately one only cares about the limit, where one has, as you write, the Black-Scholes formula.

I don't work with this, so this might be entirely academic- but isn't convergence in the binomial model extremely slow for barrier options?

It seems, after reading through the paper (to the extent my non-MIT mind understood things) that this is based upon a pricing model of European options. European options can only be exercise on the expiry date, American options can be exercised any time before that date.

I'm not sure I follow. An "American option" as you call it has two dates - one is the vesting date (the first day the option may be exercised) and an expiry date (the date the option will no longer be valid). Sometimes the vesting date can b

I'm not sure I follow. An "American option" as you call it has two dates - one is the vesting date (the first day the option may be exercised) and an expiry date (the date the option will no longer be valid).

It sounds like you are describing the options that are given out as an employment benefit. There is a different kind, traded on an exchange just like stocks are.

If a particular stock 'foo' is trading at $60 today but you think it is going to go up in the future, you can buy a "June $65 call" for some small amount of money. That option then allows you to buy a share of foo for $65 any time between the purchase date and the June expiration date (American style), regardless of what the stock price happens to b

This is the abstract used ( not really) to get teh funding grant for this research.

Two fundamentally different but complementary transition metal catalyzed chemo-, regio-,diastereo-, enantio-, and grantproposalo-selective approaches to the synthesis of a library of biologically significant nano- and pico-molecules will be presented with the focus on reaction mechanism and egocentric effects. The role of the nature of the metal, ligand, solvent, temperature, time, microwave, nanowave, picowave, ultrasound, hypersound, moon phase, and weather in this catalytic, sustainable, cost-effective, and eco-friendly technology will be discussed in detail.

I would say that the Quant methods worked very damn well. (The book "Yhe Quants" is fascinating BTW - also "Too Big to Fail" - books on CD = excellent driving amusement). In fact their effect can be seen very well as a classic 'technology bubble'. So also the 'Mortgage Bubble' where a combination of new technology (capitalization of home loans), combined with some regulatory mistakes and a large dose of people-taking-advantage on all sides.

To be more exact, there is no easy formula of the hypergeometric kind, which is a formula "involving binomial coefficients, factorials, rational functions, and power functions" according to http://mathworld.wolfram.com/HypergeometricIdentity.html [wolfram.com]. It would thus theoretically still be possible that an easy formula exists, but it must involve constructions more exotic than that.

OK, so there is no exact solution to the formula. Do you need one? Or will a Monte Carlo simulation be good enough, the way it is for (say) the physicists building nuclear bombs or the engineers designing airplanes?

Closed-form solutions are nice for proving things with arbitrary precision, but they're often not necessary in the real world, where a few decimal places often suffice.

Not to imply that the work isn't interesting. I'm sure it's got all sorts of implications with respect to the way economists analyze the algorithms. But commenters so far seem to want to jump from "no closed form exists" to "Wall Street is fundamentally unsound", which seems, uh, unsound.

More like, it's a complex subject distorted by the fact that many of the participants can make large amounts of money. And Yves is something of a rabble-rouser - her analyses are just as bad as the ones she goes to town on.

The binomial model is common in textbooks because it's intuitively appealing, but if you only apply it to basic European (exercisable at expiry) options then there really are better ways of getting a closed form solution i.e. the Black-Scholes (or Bachelier-Thorp....) formula.
If you want half decent pricing methods for more general cases then you'll end up with Finite difference or Monte-Carlo methods depending on dimensionality, at which point you've already given up on a closed form solution.
One of t

The naive CRR (Cox, Ross, Rubinstein) method for pricing options is O(n^2) where n is the number of levels in a recombinant binomial pricing lattice. That is, a lattice like a binary tree, but where you have cross links connecting nodes. The naive approach requires visiting each one of these nodes and hence O(n^2) and the error of the produced option goes down only proportional to the node spacing.
For at least 15 years this problem has been converted to "linear time" (really the important relation is between the price error and the CPU time) by means of a variety of extrapolation methods (this began with Richardson extrapolation) using evaluation with two trees to get a much smaller error. There are in fact numerical methods that for special options can do slightly better than this. Broadie 1996 is one reference. While pretty fast and very easy to understand, there are yet faster methods using adaptive mesh crank-nicolson PDE solvers that do a bit better.
Just a couple of years ago, Dai, et al. published a paper showing how to get linear time an entirely different approach involving combinatorial sums. This may have improved performance bounds for some exotic options, but did NOT do much for improving real-world implemented algorithmic performance of pricing the European and American options that are so commonly traded on exchanges, in the US and worldwide. So, at least for the most important class of options Dai et al was kind of a snoozer.
The paper referenced in the summary above is entirely a follow-up paper to Dai, et al 2008. This new paper merely shows that there is no "short cut" in evaluating the relevant sums with hypergeometric functions, a kind of special function common in mathematical physics. So, in short, all this says is that the already "non fastest method" cannot be made faster by one numerical methods approach. It is certainly deserving of publication and dissemination, but changes the world not at all.

I don't believe you understand the situation or my argument. It's not nearly as strong as P=NP or not. There could be non-hypergeometric family simplifications that do better than the Dai linear sums, and there can be other numerical methods that also do better (and there are for some kinds of options). This new paper just shows that one possible approach to simplify a formula won't work - a formula-to-be-improved already only compelling because Dai et al compare it to naive, strawmen alternatives they f

Because we have a perfectly good constitution that doesn't need to be "fundamentally transformed". We believe in the rule of law, not men, so the only "revolution" that needs to happen is to kick the lawbreakers out of our government-- and imprison them, if necessary.

I come from a country which has experienced a revolution in my lifetime.

Why can't you?

We can. We choose not to.

If your question is, "why don't you?" Then that is because I don't feel like a revolution will solve the fact that wall street is using substandard formulas (that is what we are talking about right?) and would probably cause more problems for my sound long-term investments than any pockets of casino capitalism do.

So... somebody who has no fucking clue how the real world works? You sit in your little isolated academic ivory tower and cast scorn upon the smart, hardworking, gutsy people who donated the building you work in and pay your salary - because you're too much of a coward to get a real job.

Why do Americans across the country not simply not occupy Wall Street?

Why have you been so effectively programmed to accept the shit you're fed?

Not a statement against sound long-term investment, but against casino capitalism and cronyism.

I come from a country which has experienced a revolution in my lifetime.

Why can't you?

Break and circuses. The people are well fed, well entertained, and generally very lazy. Half of the country is kept in a constant state of paranoia, hatred of gays/Muslims/liberals, and worship of the moneyed rich. They're the better armed half also. The counter revolution would nearly impossible to oppose.

It's a bit like superheated water. There's a lot of anger against banks, Wall street, and politics in general, just waiting for a nucleation point. Sort of like when you heat a cup of tea in the microwave and all seems well, then you add sugar and it explodes in a cloud of steam.

The subject is a rigorous mathematical proof that what we're told about capitalism being efficient is inherently less generally true than the sweeping, absolute terms that especially conservatives and libertarians like to claim. Direct implications of this proof include: de-regulation legislation like the Credit Futures Modernization Act & Gramm-Leach-Bliley, whose "value" was alleged to have been in facilitating more rapid exchange of capital and thereby greater "efficiency," now provably can only de

What that means is indeed absolute proof. Unless an error is found in the math, this means everything we've been told about the "benefits" that Wall Street offers society, for at least a generation, is hogwash.

You're an idiot. No, seriously, you are. The fact that you can take something you don't understand and make a sweeping statement about something else you don't understand is direct testimony to your [lack of] intelligence.

The proof is about lack of a closed form solution for an options pricing model. There are plenty of real world engineering problems for which analytical solutions do not exist and only numerical solutions work. Yet, we don't worry about planes falling out of the sky or your TV not working

The difference, and the reason your analogy is shit, is that aeronautical engineers don't claim to know the existence and smoothness of the Navier-Stokes equations in R3. If you ask them, they're honest about which of their equations are just assumptions and why they believe they're valid assumptions. What economists, especially those employed by corporate lobbying firms, have been claiming to anybody with a microphone and camera or a steno pad for decades, is that more trading is always good because, the